Question

challenge given that m∠a=(16x)°, m∠c=(7x + 30)°, and m∠d = 129°, what is m∠b? m∠b = □°

Explanation:

Step1: Recall the sum of angles in a quadrilateral

The sum of the interior angles of a quadrilateral is $360^{\circ}$. So, $m\angle A + m\angle B+m\angle C + m\angle D=360^{\circ}$.

Step2: Substitute the given angle - measures

Substitute $m\angle A=(16x)^{\circ}$, $m\angle C=(7x + 30)^{\circ}$, and $m\angle D = 129^{\circ}$ into the equation:
$16x+ m\angle B+(7x + 30)+129 = 360$.
Combine like - terms: $(16x+7x)+m\angle B+(30 + 129)=360$, which simplifies to $23x+m\angle B + 159=360$.
Then, $m\angle B=360-(23x + 159)=201 - 23x$.
We also know that for a valid quadrilateral, we first need to find the value of $x$.
Since the sum of angles in a quadrilateral is $360^{\circ}$, we have the equation:
$16x+(7x + 30)+129+m\angle B=360$.
Combining like - terms gives $23x+159+m\angle B = 360$.
We can also set up the equation using the fact that the sum of the non - known angles and the known angle must equal $360^{\circ}$.
$16x+7x+30 + 129=360 - m\angle B$.
$23x+159=360 - m\angle B$.
To find $x$, we use the angle - sum property:
$16x+7x+30+129 = 360$.
$23x+159 = 360$.
Subtract 159 from both sides:
$23x=360 - 159$.
$23x=201$.
$x = 9$.

Step3: Calculate $m\angle B$

Substitute $x = 9$ into the expression for $m\angle B$:
$m\angle B=201-23\times9$.
$m\angle B=201 - 207$.
$m\angle B = 54^{\circ}$.

Answer:

$54$